An integral–differential equation approach for the free vibration of a SDOF system with hysteretic damping

Abstract

An integral–differential equation (IDE) in the time domain is proposed for the free vibration of a single-degree-of-freedom (SDOF) system with hysteretic damping which is different from the conventional complex stiffness model as employed in the frequency domain. The integral of the Hilbert transform is embedded in the IDE and is calculated in the Cauchy principal value sense by using a numerical folding technique. Numerical experiments show that the free vibration obtained by the frequency domain approach satisfies the IDE in the time domain. A successive iteration algorithm is employed to solve the IDE subject to forced vibration, and a convergent solution for the hysteresis loop is constructed, which matches the solution found by using the frequency domain approach. Both models, the time domain and frequency domain approaches, present the noncasual effect since they are equivalent in the mathematical sense.